document Einstein - De Sitter spacetime

git-svn-id: http://svn.einsteintoolkit.org/cactus/EinsteinInitialData/Exact/trunk@59 e296648e-0e4f-0410-bd07-d597d9acff87

1 files changed, 32 insertions, 10 deletions

diff --git a/doc/documentation.tex b/doc/documentation.tex
index 7c350de..ed4fee0 100644
--- a/doc/documentation.tex
+++ b/doc/documentation.tex
@@ -90,8 +90,7 @@ parameter:
 	 Note that the parameter is {\em not\/} "Goedel"!
 	 }%%%
 	\item[{\tt "DeSitter"}]
-		De~Sitter spacetime
-		(Robertson-Walker cosmology, near $t=0$, $p=0$)
+		Einstein-De~Sitter spacetime
 	\item[{\tt "Kasner"}]
 		Kasner like spacetime
 	\item[{\tt "Milne"}]
@@ -288,11 +287,11 @@ thesis, \verb|http://www.aei.mpg.de/~jthorn/phd/html/phd.html|.)
 
 \subsection{Kerr-Schild form of boosted rotating black hole}
 
-\verb|Exact::exactmodel = "KerrSchild"| specifies Kerr spacetime
-in Kerr-Schild coordinates, as described in MTW exercise~33.8, 
-but transformed to the usual Cactus $(t,x,y,z)$ Cartesian-topology
-coordinates, and Lorentz boosted in the $z$ direction so the black hole
-is centered at the position $z = vt$.  The physics parameters are
+\verb|Exact::exactmodel = "KerrSchild"| specifies Kerr spacetime in
+Kerr-Schild coordinates, as described in MTW exercise~33.8, Lorentz
+boosted in the $z$ direction so the black hole is centered at the
+position $z = vt$, and transformed to the usual Cactus $(t,x,y,z)$
+Cartesian-topology coordinates.  The physics parameters are
 \begin{align}
 a & = \text{\tt KerrSchild\_a}						\\
 m & = \text{\tt KerrSchild\_m}						\\
@@ -333,8 +332,8 @@ is a null vector.
 
 \subsection{Kerr spacetime in cartesian coordinates}
 
-\verb|Exact::exactmodel = "KerrSchild"| specifies Kerr spacetime
-in XXX coordinates.
+\verb|Exact::exactmodel = "Kerr"| specifies Kerr spacetime in
+FIXME-WHAT-IS-TIME-SLICING coordinates.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -358,7 +357,27 @@ in XXX coordinates.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\subsection{De~Sitter spacetime (Robertson-Walker cosmology, near $t=0$, $p=0$)}
+\subsection{De~Sitter spacetime}
+
+\verb|Exact::exactmodel = "DeSitter"| specifies Einstein-De~Sitter
+spacetime, as described in Hawking and Ellis section~5.3 and MTW
+section~27.11, transformed to the usual Cactus $(t,x,y,z)$
+Cartesian-topology coordinates.  The only physics parameter is
+\begin{equation}
+a = \verb|Desitt_a|
+\end{equation}
+
+The Einstein-De~Sitter line element in $(t,r,\theta,\phi)$ coordinates
+is given by
+\begin{equation}
+ds^2 = -dt^2 + a t^{4/3} \left[ dr^2 + r^2 d\Omega^2 \right]
+\end{equation}
+and is a special case (flat spatial geometry, no pressure) of the more
+general Robertson-Walker metric.  The only non-vanishing component of
+the stress-energy tensor is
+\begin{equation}
+T_{tt} = \frac{1}{6 \pi t^2}
+\end{equation}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
@@ -393,6 +412,9 @@ in XXX coordinates.
 \section{Acknowledgments}
 
 Many, many people have contributed code to this thorn.
+Jonathan Thornburg wrote this documentation in May 2002 based on
+the comments in the code, some reverse-engineering, and querying
+various people about how the code works.
 
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